TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-03

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A module homomorphism vanishing on $N$ factors uniquely through $M / N$

Statement

Let $f : M \to P$ be a module homomorphism and let $N \leq M$ satisfy $N \subseteq \ker f$ . There is a unique module homomorphism

$\overset{ˉ}{f} : M / N ⟶ P$

such that $\overset{ˉ}{f} (m + N) = f (m)$ , equivalently $f = \overset{ˉ}{f} \circ π$ .

Facts & Assumptions

Given: A module homomorphism $f : M \to P$ and a submodule $N \leq M$ with $N \subseteq \ker f$ .

[L1]

The canonical map $π : M \to M / N$ is a surjective module homomorphism with kernel $N$ (The canonical map $M \to M / N$ is a surjective module homomorphism with kernel $N$ ; thus every submodule is a kernel).

[L2]

A group homomorphism that kills a normal subgroup factors uniquely through the group quotient (A homomorphism that kills a normal subgroup factors uniquely through the quotient group).

[L3]

A module homomorphism is additive and scalar-preserving, and its kernel is the preimage of zero (Module homomorphism and isomorphism, kernel, image and cokernel).

Proof

technique · constructive

1.1

By [L1], the canonical map $π : M \to M / N$ is an additive-group quotient map with kernel $N$ . Viewing the modules as additive groups, [L3] makes $f$ a group homomorphism and the hypothesis says it kills $N$ . By [L2], define the unique additive homomorphism $\overset{ˉ}{f}$ by $\overset{ˉ}{f} (m + N) = f (m)$ , with $f = \overset{ˉ}{f} \circ π$ .

L1L2L3givenconstruct

2.1

For every coset, $\overset{ˉ}{f} (r (m + N)) = \overset{ˉ}{f} (r m + N) = f (r m) = r f (m) = r \overset{ˉ}{f} (m + N)$ , so $\overset{ˉ}{f}$ is scalar-preserving.

step 1.1L3given

3.1

Thus $\overset{ˉ}{f}$ is a module homomorphism with the required factorisation.

step 1.1step 2.1L3

4.1

Any module-homomorphism factor is in particular an additive-group factor, so the uniqueness in step 1.1 proves its uniqueness as a module homomorphism.

step 1.1L2L3discharge-construct: final∎

Depends on

Used by

M⊗_RR/I≅ M/IM naturally Corollary
Symmetric and exterior powers are representations Proposition
Tangent space of a free proper quotient Proposition
Tangent space of a homogeneous quotient Proposition
A module-valued coend is the direct sum of the diagonal values modulo the dinaturality submodule Theorem
Correspondence theorem for submodules of a quotient module Theorem
Covariant and contravariant Hom are left exact Theorem
First isomorphism theorem for modules: M/ker f congimf Theorem
For every ring R, the category R-Mod is complete and cocomplete Theorem
Tensoring is right exact Theorem
The Snake Lemma for modules Theorem
Third isomorphism theorem for modules Theorem

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)

A module homomorphism vanishing on NN factors uniquely through M/NM/N